Monday, April 26, 2010
Sunday, April 25, 2010
Friday, April 23, 2010
Monday, April 19, 2010
Ironically, I've found much less material about Infinity than about Nothing (though a lot of the stuff about Nothing talks about Infinity, too). But perhaps that's paradoxically appropriate.
Mystery has its own mysteries, and there are gods above gods. We have ours, they have theirs. That is what's known as infinity.
—Jean Cocteau (1889-1963), French author and filmmaker
A concept that has always fascinated philosophers and theologians, linked as it is to the notions of unending distance or space, eternity, and God, but that was avoided or met with open hostility throughout most of the history of mathematics. Only within the past century or so have mathematicians dealt with it head on and accepted infinity as a number — albeit the strangest one we know.
An early glimpse of the perils of the infinite came to Zeno of Elae through his paradoxes, the best known of which pits Achilles in a race against a tortoise. Confident of victory, Achilles gives the tortoise a head start. But then how can he ever overtake the sluggish reptile? asks Zeno. First he must catch up to the point where it began, by which time the tortoise will have moved on. When he makes up the new distance that separated them, he finds his adversary has advanced again. And so it goes on, indefinitely. No matter how many times Achilles reaches the point where his competitor was, the tortoise has progressed a bit further. So perplexed was Zeno by this problem that he decided not only was it best to avoid thinking about the infinite but also that motion was impossible! A similar shock lay in store for Pythagoras and his followers who were convinced that everything in the universe could ultimately be understood in terms of whole numbers (even common fractions being just one whole number divided by another). The square root of 2 — the length of the diagonal of a right-angled triangle whose shorter sides are both one unit long — refused to fit into this neat cosmic scheme. It was an irrational number, inexpressible as the ratio of two integers. Put another way, its decimal expansion goes on forever without ever settling into a recurring pattern.
These two examples highlight the basic problem in coming to grips with infinity. Our imaginations can cope with something that hasn't yet reached an end: we can always picture taking another step, adding one more to a total, or visualizing another term in a long series. But infinity, taken as a whole, boggles the mind. For mathematicians this was a particularly serious problem because mathematics deals with precise quantities and meticulously well-defined concepts. How could they work with things that clearly existed and went on indefinitely — a number like sqrt(2) or a curve that approached a line ever more closely — while avoiding a confrontation with infinity itself? Aristotle provided the key by arguing that there were two kinds of infinity. Actual infinity, or completed infinity, which he believed could not exist, is endlessness fully realized at some point in time. Potential infinity, which Aristotle insisted was manifest in nature — for example, in the unending cycle of the seasons or the indefinite divisibility of a piece of gold — is infinitude spread over unlimited time. This fundamental distinction persisted in mathematics for more than 2,000 years. In 1831 no less a figure than Karl Gauss expressed his "horror of the actual infinitude," saying:
I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.
By confining their attention to potential infinity, mathematicians were able to address and develop crucial concepts such as those of infinite series, limit, and infinitesimals, and so arrive at the calculus, without having to grant that infinity itself was a mathematical object. Yet as early as the Middle Ages certain paradoxes and puzzles arose, which suggested that actual infinity was not an issue to be easily dismissed. These puzzles stem from the principle that it is possible to pair off, or put in one-to-one correspondence, all the members of one collection of objects with all those of another of equal size. Applied to indefinitely large collections, however, this principle seemed to flout a commonsense idea first expressed by
Imagine, said Hilbert, a hotel with an infinite number of rooms. In the usual kind of hotel, with finite accommodation, no more guests can be squeezed in once all the rooms are full. But "Hilbert's Grand Hotel" is dramatically different. If the guest occupying room 1 moves to room 2, the occupant of room 2 moves to room 3, and so on, all the way down the line, a newcomer can be placed in room 1. In fact, space can be made for an infinite number of new clients by moving the occupants of rooms 1, 2, 3, etc, to rooms 2, 4, 6, etc, thus freeing up all the odd-numbered rooms. Even if an infinite number of coaches were to arrive each carrying an infinite number of passengers, no one would have to be turned away: first the odd-numbered rooms would be emptied as above, then the first coach's load would be put in rooms 3n for n = 1, 2, 3, ..., the second coach's load in rooms 5n for n = 1, 2, ..., and so on; in general, the people aboard coach number i would empty into rooms pn where p is the (i+1)th prime number.
Such is the looking-glass world that opens up once the reality of sets of numbers with infinitely many elements is accepted. That was a crucial issue facing mathematicians in the late nineteenth century: Were they prepared to embrace actual infinity as a number? Most were still aligned with Aristotle and Gauss in opposing the idea. But a few, including Richard Dedekind and, above all, Georg Cantor, realized that the time had come to put the concept of infinite sets on a firm logical foundation.
Cantor accepted that the well-known pairing-off principle, used to determine if two finite sets are equal, is just as applicable to infinite sets. It followed that there really are just as many even positive integers as there are positive integers altogether. This was no paradox, he realized, but the defining property of infinite sets: the whole is no bigger than some of its parts. He went on to show that the set of all positive integers, 1, 2, 3, ..., contains precisely as many members — that is, has the same cardinal number or cardinality — as the set of all rational numbers (numbers that can be written in the form p/q, where p and q are integers). He called this infinite cardinal number aleph-null, "aleph" being the first letter of the Hebrew alphabet. He then demonstrated, using what has become known as Cantor's theorem, that there is a hierarchy of infinities of which aleph-null is the smallest. Essentially, he proved that the cardinal number of all the subsets — the different ways of arranging the elements — of a set of size aleph-null is a bigger form of infinity, which he called aleph-one. Similarly, the cardinality of the set of subsets of aleph-one is a still bigger infinity, known as aleph-two. And so on, indefinitely, leading to an infinite number of different infinities.
Cantor believed that aleph-one was identical with the total number of mathematical points on a line, which, astonishingly, he found was the same as the number of points on a plane or in any higher n-dimensional space. This infinity of spatial points, known as the power of the continuum, c, is the set of all real numbers (all rational numbers plus all irrational numbers). Cantor's continuum hypothesis asserts that c = aleph-one, which is equivalent to saying that there is no infinite set with a cardinality between that of the integers and the reals. Yet, despite much effort, Cantor was never able to prove or disprove his continuum hypothesis. We now know why — and it strikes to the very foundations of mathematics.
In the 1930s, Kurt Gödel showed that it is impossible to disprove the continuum hypothesis from the standard axioms of set theory. Three decades later, Paul Cohen showed that it cannot be proven from those same axioms either. Such a situation had been on the cards ever since the emergence of Gödel's incompleteness theorem. But the independence of the continuum hypothesis was still unsettling because it was the first concrete example of an important question that provably could not be decided either way from the universally-accepted system of axioms on which most of mathematics is built.
Currently, the preference among mathematicians is to regard the Continuum Hypothesis as being false, simply because of the usefulness of the results that can be derived this way. As for the nature of the various types of infinities and the very existence of infinite sets, these depend crucially on what number theory is being used.
Different axioms and rules lead to different answers to the question what lies beyond all the integers? This can make it difficult or even meaningless to compare the various types of infinities that arise and to determine their relative size, although within any given number system the infinities can usually be put into a clear order. Certain extended number systems, such as the surreal numbers, incorporate both the ordinary (finite) numbers and a diversity of infinite numbers. However, whatever number system is chosen, there will inevitably be inaccessible infinities — infinities that are larger than any of those the system is capable of producing.
The Infinite Book by John D. Barrow
The best and most readable primer on the subject by the author of The Book of Nothing.
Achilles in the Quantum Universe by Richard Morris
The first book I read on the the subject, this is another very good and readable account of the history and use of Infinity.
Infinity: The Quest to Think the Unthinkable by Brian Clegg
Another overview, this one isn't as good as The Infinite Book and Achilles, but it's worth checking out.
A Brief History of Infinity by Paolo Zellini
Infinity and the Mind: The Science and Philosophy of the Infinite by Rudy Rucker
These are books I've not actually read, but for the sake of completeness have on my list.
Sunday, April 18, 2010
Friday, April 16, 2010
I never stood a chance, really; my dad was a Trekkie from TOS days and I'm distressingly like him, so it was inevitable. (Though I didn't get his Western fixation.)
Tuesday, April 13, 2010
Sunday, April 11, 2010
Thursday, April 08, 2010
Wednesday, April 07, 2010
Tuesday, April 06, 2010
Monday, April 05, 2010
Sunday, April 04, 2010
The Origin of Zero
A very brief history of that number.
Being in Nothingness
Random New Yorkers talk about nothing.
Why Is There Something Rather Than Nothing?
One of the biggest arguments people have for God is the existence of... existence, rather than nonexistence. It isn't as strong an argument as some think.
Writing home about nothing
An interview with Frank Close, author of The Void and Nothing: A Very Short Introduction. To add an audiovisual component: Frank Close Talks About Nothing.
The Importance of Being Nothingness
Review of two of my Books About Nothing.
A philosopher muses on Nothing.
Nothingness (Standford Encyclopedia of Philosophy) & Nothing (The Encyclopedia of Philosophy)
Articles providing much more in-depth treatment of the subject as it pertains to philosophy.
Best popular article I've found dealing with Nothing. A must-read.
Nothing is as symmetrical as Nothing.
Thursday, April 01, 2010
This is the book that kindled my interest in Nothing. Before that, I'd never really contemplating the subject, but once my eyes were open there was no going back.
The book is concerned, as you might guess, with the number zero, its history and use. Seife is a terrific author, and the book is a very easy, fun read; he really emphasizes the cultural aspects of zero alongside the math. It's a great introduction to the subject.
The Book of Nothing by John Barrow
This book is more expansive than Zero, and expands more into other aspects of Nothing like physics and art. John Barrow is one of my favorite science writers, and this book is what turned me on to him.
The Hole in the Universe by K.C. Cole
This book mainly deals with physics, but also gets into psychology and cognitive science and revels in the linguistic games nothing plays.
The Nothing That Is: A Natural History of Zero by Robert Kaplan
Another book tracing the origins and usage of zero, it is not as interesting or wide-ranging as Seife's book. It's much more focused on the mathematics.
The Void & Nothing: A Very Short Introduction by Frank Close
The most recent full treatments on Nothing.
Nothingness: The Science of Empty Space by Henning Genz
A good overview of Nothing.
Patterns in the Void: Why Nothing Is Important by Sten Odenwald
Probably my favorite book about Nothing, because it uses Odenwald's own life and relationship with Nothing to illustrate the concepts. It's just a really enjoyable, well-written book.
Signifying Nothing: The Semiotics of Zero by Brian Rotman
This is a book I don't have and have only really ever flipped through in the library, but for completeness sake I list it here. It's not on a subject I find terribly interesting, but perhaps some day I'll look at it again and find I was wrong, or maybe you'll just be more interested.
You Don't Have To Be Buddhist To Know Nothing by Joan Konner (ed.)
The latest addition to my library of nothing, this is a compendium of quotes from a wide range of notables -- from artists to poets to philosophers to scientists and beyond -- about Nothing. Several of the quotes the previous post are garnered from it. It is also the impetus for this series.